Schemes for Deterministic Polynomial Factoring

TL;DR

This paper introduces m-schemes to connect the deterministic complexity of polynomial factoring over finite fields with combinatorial objects, leading to a new polynomial-time algorithm under GRH for certain prime degree polynomials.

Contribution

It extends the deterministic subexponential algorithm for finite field polynomial factoring by incorporating m-schemes, achieving a polynomial-time algorithm under GRH for specific prime degrees.

Findings

Established a link between m-schemes and polynomial factoring complexity

Developed a deterministic polynomial-time algorithm under GRH for prime degree polynomials with smooth (n-1)

Demonstrated the effectiveness of m-schemes in improving factoring algorithms

Abstract

In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring algorithm for finite fields to get an underlying m-scheme. We demonstrate how the properties of m-schemes relate to improvements in the deterministic complexity of factoring polynomials over finite fields assuming the generalized Riemann Hypothesis (GRH). In particular, we give the first deterministic polynomial time algorithm (assuming GRH) to find a nontrivial factor of a polynomial of prime degree n where (n-1) is a smooth number.